184 Mh MURPHY'S SECOND MEMOIR ON THE 



Cor. 1. The simplest form for a„ is the sine or cosine of the w'" 

 multiple of an arc of which the limits are and 2w7r, as 



A„ sin (2 nicT) + B„ cos (2 wtt T), 



where A„, B„ are arbitrary constants, then we have (putting for sim- 

 plicity a = 0), 



S„ = A,P„ + {A, sin ^TTT + B, cos ^-n-r) -^ 



+ {Ai sin 4 TTT + Bi cos 4 ttt) 



dt 



■i > 



this is the most general form for all the reciprocal functions which occur 

 in the Mecanique Celeste. (Vid. Prop, xi. Treatise on Electricity.) 



CoK. 2. If T„, T,' are arbitrary functions of t, which do not become 

 infinite when ^=0 or 1, then, putting 



Rn = {tt'f Tr*^, and R,: = {tt'f T; .^ , 



the same reasoning as that used in the preceding example will show 

 that R^, R„' are reciprocal functions, and thus we get for a^^, aS",,' the 

 very general forms 



S„ = «„ T,P„ + «. y. ^ («')* + «^ T^ -^ m + «3 T, ^ {tt'f + &c. 



S: = a„' 2;'P„ + a/ T; "^ {tt'f + a.: T^ ^ {tt') + ai Ti ^{tt'f + &C. 



Cor. 3. If f{t, t) is any function of the variables /, t, which is ex- 

 panded under the form 



f{t, t) = a,S^ + a,Si + a^S; + 



then, to determine the coefficients a^, Ui, a-i, &c., multiply successively 

 by So, Si, SJ.... and integrate from t=0 to t=l, and from t = to 

 T = 1 : we thus get 



do ft fr So So = ftfTSo'J'{t, t), 



aiftfrSiSi' = ftf,Si/{t, t), 

 aJJ^S.,S.; = f,f^S./f{t,T); 

 from whence the required coefficients are known. 



